Some Finite Abelian Group Theory and Some q-Series Identities
Annals of Combinatorics
The purpose of this article is to compute certain weighted sums over various subposets of the poset of isomorphism classes of finite abelian ℓℓ-groups, where ℓℓ is a fixed odd prime. These sums are similar to previous sums computed by Hall and Cohen-Lenstra. The computation expands upon the previous analysis of Cohen-Lenstra while also using the tools of hypergeometric functions and qq-series. The identities computed in this article, while aesthetically pleasing in their own right, also turn out to be applicable to the construction of a random matrix version of the Cohen-Lenstra-Martinet heuristics.
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Garton, D.Some Finite Abelian Group Theory and Some q-Series Identities. Annals of Combinatorics. (2016) 20: 361.